Data Structure for Real-Time Processing in 3-D

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Data Structure for Real-Time

Processing in 3-D

Jean-François Lalonde, Nicolas Vandapel

and Martial Hebert

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Problem

Dynamic processing of large 3-D point cloud data from

ladar

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Example

• Terrain classification

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Local computation on 3-D point sets

Point of interest

Scan through all points in the dataset

Support region

Local computation on highlighted points

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Local computation on 3-D point sets

Point of interest

Scan through all points in the dataset

Support region

Local computation on highlighted points

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• Compute scatter matrix within support volume

• Extract principal components

• Features are linear combination of eigenvalues

[Tang-PAMI04]

• Voxelize data

• Store sufficient statistics for scatter matrix in voxels

– Sums, sums of squared and sums of cross-products of 3-D points

coordinates

– Minimize storage, reduce amount of data without losing

information for later processing

Local computation on 3-D point sets:

example

2 1 0 λ λ λ ≈ ≈ 0 scatter F =λ 2 λ 1 λ 0 λ 2 1 0 λ λ λ ≈ > > ( 1 2) 2 planar F = λ −λ ⋅e 0 λ 2 λ 1 λ 2 1 0 λ λ λ > > ≈ ( 0 1) 0 linear F = λ −λ ⋅e

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Challenges

• Nature of data

– Ladar on a moving platform [Lacaze-AUVSI02]

• Dynamic (accumulation)

– Need to process data continuously

• Efficient operations

– Insertion and access

– Range search

• Local computations

• Traditional techniques do not apply

– Tree-based data structures [Samet81, Liu-NIPS04,

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Concept – 2-D example

Voxel

Stores sufficient statistics of all points

that fall in it

Support region

Sum sufficient statistics of all voxels within

k

k = 5

Size of support region

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Concept – 2-D example

Overlap

How can we reuse pre-computed data?

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Concept – 2-D example

1. Start with the blue region

2. Add the green column

3. Subtract the red column

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Concept – 2-D example

• Proven to be efficient in image processing [Faugeras93]

• Challenge in 3-D: data is sparse

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2-D example, sparse data

Empty voxel

Sparse data

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2-D example, sparse data

May not always be

useful to reuse data

Empty voxel

1. Start with the blue region

2. Add the green columns

3. Subtract the red columns

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2-D example, sparse data

2 approaches:

2. Default scan

Where is the

previous result?

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Approach 1: default scan

x y

k = 5

Size of support region (in # of voxels) k

2. Memory

Compute partial sums

1. Scanning

direction

Example: x first Arbitrary

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Approach 1: default scan

x y

k = 5

Size of support region (in # of voxels) k

2. Memory

Compute partial sums

1. Scanning

direction

Example: x first Arbitrary d

d = 2

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2 cases

d k d k

2 k

d

<

2 k

d

>

d = 3

d = 2

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Approach 2: optimized scan

• Can we do better?

x y

Would be better to

choose the result from

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Approach 2: optimized scan

Empty voxel

Additional arrays

Store all previous results & locations

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Approach 2: optimized scan

Empty voxel

Additional arrays

d_min

d

_min

Distance between voxel of interest and closest

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Approach 2: optimized scan

Empty voxel

Additional arrays

d_min

d

_min

Distance between voxel of interest and closest

previous result

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Comparison

+ Very easy to implement + Minimal overhead

one memory location

one distance computation

+ Independent on scanning direction + Provide highest speedup

- Harder to implement

direction determined dynamically

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Experiments - overview

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Experiments - overview

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Experiments - overview

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Experiments - overview

Forest dataset

Flat ground dataset Tall grass dataset

• Voxel size of 0.1m

• Experiments:

– Influence of scanning direction

– Speedup on different scenes

– Influence of data density

117,000 occupied voxels 112,000 occupied voxels

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Experiments – scanning direction

Flat ground dataset

y z y z x y z

Optimized version Along x

Along y Along z Distance to previous occupied voxel Distance to previous occupied voxel F re q u e n cy F re q u e n cy F re q u e n cy F re q u e n cy Avg. dist = 1.15 Freq = 99% Avg. dist = 1.75 Freq = 94% Avg. dist = 1.79 Freq = 96% Avg. dist = 1.12 Freq = 64%

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Experiments – scanning direction

Forest dataset

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Experiments - speedup

Forest dataset

T im e, n o rm al iz ed ( m s/ v o x el )

Radius of support region (m)

Direct computation Direct computation Direct computation

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Experiments - density

Tall grass dataset

Old method, Direct computation

New method, Optimized scan

Raw

Radius of support region (m)

T im e, n o rm a li ze d ( m s/ v o x e l) Sub-sampled Raw Sub-sampled Nb of voxels Raw: 117,000 Sub-sampled: 9,000

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What can we predict?

v

Number of occupied voxels

n

Total number of

voxels in the volume

_{P[X < k/2]}

Probability that condition for reuse is satisfied

k

Size of support region

d

Average distance between interest voxel

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Experimental validation

0 0,02 0,04 0,06 0,08 0,1 0,12 0 0,005 0,01 0,015 0,02 0,025 T im e, n o rm a li ze d ( m s/ v o x e l) Volume occupancy (%)